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JOURNAL OF ECONOMIC Growth THEORY and Optimal 10, 239-257 (1975) Intertemporal JEAN-MICHEL Ingknieur au Corps Allocation of Risks* BISMUT des Mines Received December 17. 1974 It has been said that capital theory is the theory of time, because its concern is to explain how a given amount of resources should be allocated in time to reach a certain optimum, which may be the maximization of profit for a firm, or the maximum of an intertemporal welfare function for the whole economy. We would like to show here how the introduction of uncertainty can modify the classical results of optimal growth theory, not to develop a mathematical apparatus, which can be cumbersome, but to understand the underlying economic concepts which appear in the mathematical theory, and to interpret, in economic terms, the theoretic tools which are used in the mathematical theory. We want in particular to understand how time and risk are connected. It is obvious that in an accumulation process with uncertainty, decisions have uncertain consequences. This implies that a given investment decision at time t has an impact not only on the expected value of capital at time t + dt, but also on all its probability distributions-for instance on its variance. But the problem is that when one wants to maximize the expected value of a given criteria, there is in a loose sense an “addition” of risks. The optimal intertemporal allocations of resources will then necessarily lead to an optimal intertemporal allocation of risks. At the same time, information plays a basic role in the problem, in the sense that one may differ a given decision of investment because of lack of information on what the expected technological changes are going to be. Although we do not suppose that the investor controls the information he receives, there is, through the use he makes of the available information, an optimal intertemporal utilization of information. We are then concerned here with two essential aspects of intertemporal decision making under uncertainty: risk-taking and information processing. * The author is indebted to R. C. Merton for very helpful discussions’and referees for their very useful comments and suggestions. Copytight AU rights 0 1975 by Academic Press, Inc. of reproduction in any form reserved. 239 to two 240 JEAN-MICHEL BISMUT The introduction of dual variables will be particularly helpful to treat the problem. This paper relies heavily on previous mathematical work by the author. In [2] and [3], we have developped a new approach to problems of optimal stochastic control, which we will interpret here intuitively. We will not try to be rigorous in the mathematics used here. The reader interested in the mathematical aspects of the problem is referred to [2] and [3]. I. THE EQUATIONS OF THE MODEL We assume that a firm is accumulating capital k through investment decisions U. At the same time environmental factors-such as technological changes, stock market fluctuations-have a direct action on the accumulation process. We assume that the investment decisions and the environmental factors modify the infinitesimal increments of capital stock through two factors, characterizing its probability distributions, namely its meanf, and its variance (T. This simplification has already been strongly utilized for stock-market models. However, due to its considerable intuitive appeal and to its technical simple handling, we use it here. Formally, /I being a m-dimensional Brownian motion1 (that is a stationary continuous process with independent Gaussian increments) the accumulation equation is: dk = f(u, t, k, u) dt + a(w, t, k, u) * d/3, (l-1) where : k is the capital stock, u is the investment decision, w is the environmental factor. k will then have continuous trajectories. This is still compatible with the idea of a capital stock increasing continuously. Besides, it is quite natural to say that mean and variance of the infinitesimal increment depend on the size of the capital stock as well as on the environmental factors and on the investment decisions. In particular, a given investment decision may involve more or less risks according to the size of the accumulated capital stock. We assume also that the investor has an increasing system of information {Fth,, 3 that his information includes the past values of k and /3, and 1 It is not essential that ,8 is a Brownian motion. It would be sufficient that j? is any martingale such that one can find a& = I,..., m) such that pia - J-i a6 dt is a martingale. q is interpreted as variance only for simplification. GROWTH AND OPTIMAL INTERTEMPORAL ALLOCATION 241 also f(q t, ., .) and a(@, f, *, a): he knows then the expected mean and variance of the capital increment consecutive to any decision he makes. His goal is to maximize the expected value of total profits up to a certain time T which we write: E j’ 0 L(w, t, k, u) dt, (l-2) L(w, t, k, u) dt being the instantaneous profit, which is also supposed to be known by the investor. Instantaneous risk aversion or time preferences are supposed to be taken into account in the criteria. We have now a difficult intertemporal optimization problem with uncertainty to solve. Let us represent in Figure 1 a simplified representation of the decision process. As time elapses, the investor must take into account the new information he receives, and using it, evaluate the influence on the expected pay-off of his decision. FIGURE 1 II. TIME DECENTRALIZATION We know that in the deterministic case, the introduction of a dual variable enables us to change an intertemporal maximization problem to an instantaneous maximization problem, that is, we can do a time decentralization of decisions.2 2 For a rigorous exposition, the reader is referred to 131. 242 JEAN-MICHEL BISMUT Although what we will do here is not rigorous, let us figure out what a time decentralization can be when uncertainty is present. Let pt be the marginal value of capital at time t, that is, the partial “derivative” with respect to k of the conditional expectation of the optimum future profits from time f: aEFt T Pt = __i3k j t L(w t, k 4 dt (2.1) u being an optimal policy. We assume that pt may be written: !‘t=Po+ jotAds+ jotIVdB,+Mt (2.2) where fi8 is the infinitesimal expected rate of growth of p, H, is the infinitesimal conditional covariance of p with /3, M is a predictive term, null at 0, which is the best estimate at time t of a given random variable, and which we suppose to be in a loose sense independent of 8. Its infinitesimal increments have then a null conditional expected value at each time. This decomposition is not unnatural. It corresponds to the idea that p can be decomposed in the sum of p. , of a term Ji J’Jwhich gives its expected infinitesimal increment at each time, a term which integrates uncertainties on the accumulation process itself and finally a term integrating the information on environmental factorsMt . Then at time t, it is intuitively appealing to write that the optimal trajectory maximizes at any time t the sum of the expected infinitesimal future profits and of the expected increment in the value of the capital stock, that is: EFt jtiida L(w, t, k, u) dt + EFt f+at {jk + EF’ jt- + pf(o, s, k, 1.41ds <H, +J, s, k, 41 ds. (This calculation follows from Ito rules on stochastic calculus.) (2.3) GROWTH AN0 OPTIMAL INTERTEMPORAL ALLOCATION 243 This is equivalent to saying that L(w, t, k 4 + #k + pf(w, s, k, 4 + W, &J, s, k 4) (2.4) is maximal on the optimal trajectory. We define Z by: 8 = GJ, t, k, 4 + ~f(w, s, k, u) + W, 4~ s, k, 4) (2.5) We have then the following relations for u and p: as/au = 0. dp = --(asyxj dt + H * dp + div. (2.6) pT = 0. These equations are precisely the equation given for the first time in [2, 31. We see then how the introduction of the risk term u - d/l modifies the classical Pontryagin principle by obliging the investor to take into account the instantaneously uncertain consequences of his action: this corresponds to the term H. In the same way, the uncertainty on the environmental factors obliges the investor to do some prediction through M. Roughly speaking, we have the following correspondence of primal and dual quantities. f--P. u --f H. (2.7) Ft - Aft . III. INTERPRETATION We now try to interpret the different quantities appearing in Part II. We consider first H and &‘. With the previous reasoning, and by analogy with stock-market analysis, it appears immediately that -H can be interpreted as the cost of risk taking at time t.S Then, 2 is the sum of the instantaneous profit L, of the expected infinitesimal increment of capital valued at its marginal expected value, minus the risk associated to a given investment policy, valued at its cost. a There are obviously some dimension problems, because H is homogeneous to a We do not insist on them. price/fx~x. 244 JEAN-MICHEL BISMUT In any case, His an unknown of the problem, and is determined through the system [Eq. (2.6)J. Figure 2 represents the considered maximization. Ltpf FIGURE 2 We then maximize in R2 the quantity E + Ho, where (E, H) is (L(w, t, k, U) + pf(w, t, k, u), a(~, t, k, u)) when u describes the control set. The line has a slope which is -H. We have then changed the global optimization problem into a sequence of selection of “efficient points” in a (expected value, standard deviation) region. We now interpret the second part of system (2.6). It means first that the conditional expected rate of depreciation of one marginal unit of capital, which is -a, is equal to the sum of its contribution to the instantaneous profits, plus its contribution to enhancing the expected value of the increment of the capital stock, minus its contribution to increasing the conditional standard deviation of the increment of the capital stock valued at the cost of risk. In other words, knowing the instantaneous attitude toward risk, given by H-which is positive if the investor is risk taking and negative if he is risk averting, the investor is able to weight present certain profits and future uncertain profits, and then to compute the expected rate of depreciation. If we interpret -$ as the expected loss incurred if the acquisition of a unit of capital is postponed, it is then natural that this loss depends on the result of random choice d/3, through the standard deviation cr, because the postponment of the acquisition of a unit of capital has anyway uncertain consequences at time t + dt. We compare then the sure acquisition of a unit of capital now at time t, which will increase the stock of capital at time t + dt of an uncertain quantity versus the sure acquisition of a unit of capital at time t + dt. The uncertainty between t and t + dt tends to reduce the rate of depreciation because of the acquisition of information between t and t + dt (the “positivity” of this reduction depends on the sign of -H, which will be positive if the investor is instantaneously risk averter). GROWTH AND OPTIMAL INTERTEMPORAL 245 ALLOCATION We see here how the irreversibility of the accumulation process modifies the inverstor’s attitude. We must now interpret the remaining terms in the expression of dp. As we have seen, H dt represents the conditional infinitesimal covariance of dp and d/3. We can write: d/3 = (l/u)(dk --fdt). (3.1) Then: H dp = (H/u)(dk --fdt). (3.2) The term H d/3 is then a correction term in the evolution of p, which evaluates in terms of p the difference between dk and E(dk) which isfdt. Finally, the term A4 integrates the necessary information, which, loosely speaking, is not contained in the past values of /I. In particular, the information may come discontinuously, for instance because of unpredictable changes in the political situation, or because of fire in a plant, or because **. the model of stock market prices taken by the investor involves discontinuous changes. At these times, the investor must necessarily reevaluate his predictions, to take into account the new information which is now available. The integration of this new information is done precisely through the term dM. New information can then either significantly raise the marginal value of capital, or lower it. To put things more briefly, although more imprecisely, if we admit that /3 contains all the short-term uncertainties which appear in the capital accumulation process, M is a prediction on the long term uncertainties. Even if the continuous time model can be criticized for lack of concreteness, it appears that the interpretation of M is intuitively quite appealing, as it allows us to represent formally in the decision-making process the part of the information processing, which can be done continuously, that is, on a “routine basis,” and the part which is done discontinuously, that is, by reevaluating the predictions. k I’ i, expected rate of depreciation t+dt FIGURE 3 246 JEAN-MICHEL BISMUT In the terminology of Fama in [6], the processp, which can be interpreted as a price process, satisfies the semistrong form tests because the price system integrates all the information available to the investor. Figure 3 represents the dual system. IV. SOME EXAMPLES 1. A Stock-Market Model We take here first the model given by Merton in [7]. The investor’s wealth is represented by W and C is his personal consumption. He invests in stocks, whose prices are supposed to be led by an equation of the type: dPi/Pi = rx,(Pi , t) dt + q(Pi , t) dpi , (4.1) where 13, (i = 1 .. . n) is an n-dimensional Brownian motion (the independence of the & simplifies slightly the model) and where oi > 0 (i = 1 . .. n). If r is the market interest rate, and wi is the proportion of wealth invested in asset i, 1 - xy=, wi is the proportion of wealth invested in the riskless asset, the rate of return of which is r. The budget equation is then (see [7] (14’)): dW=fwi(ui-r) Wdt+(rW-C)dt+fwiWui&, 1 1 (4.2) W(0) = wg . The objective of the investor is to maximize: E j. T U(C, t) dt + B( W(T)). (4.3) 0 We write the generalized Hamiltonian X= S: U(C,t)+p +~H,w~WCT~. 1 If we maximize ~‘2’ in C and wi , we have: v’(C, t> = p, (PC% - r) 4 H&I W = 0. GROWTH AND OPTIMAL INTERTEMPORAL 247 ALLOCATION We have then Hi = -p(q - l-)/q . (4.4) The equation for p is then: i: Wi(% - dp = --P r) + r + f 1 y)dt + f Hi .&4 (4.5) 1 1 or: dp = -pr dt - Cn p(“iur Integration r, dpi . I 1 of (4.6) gives: -rt f-5 = p. ew _ i $ St (ai uy2r)2 dt _ ‘f 1” 7 (4 0 z 10 d/ji/, (4.7) ' p. being a constant such that PT = @jaw> (4.8) ww. (If W, = 0 there is no condition on pT .) We deduce immediately that for any kind of investor, all the processesp, are proportional. The expected marginal values of wealth for all investors are then all proportional. Moreover, the relation uyc, t) = p proves that at the optimum, all the consumption processes are functions of the same process given by formula (4.7) with p. = 1. In the case where 01~and ui are constant-they do not depend on Pi, the i>rocess p will then be log normal. It is a remarkable feature that when ail the asset prices are supposed to be log-normal, the marginal utility of wealth is also a log-normal process (we had already said that p has some of the features of a price). If p: is the process exp I -rt - i n (ai - r)2 C 1 4(ypit/ ui2 I1 (4.10) in the notations of [6] if J is defined by: 4 There is no term M, because no environmental factors are modifying the system. 248 JEAN-MICHEL BISMUT then: JW=p, (4.12) Then generally W can be written as: K = WPtO, t>* (4.13) Formula (30) of L6] implies that wi can be written: %(W, 0 = hi + m(W, ggi, (4.14) hi and gi not depending on U. This implies that W, m, wi , C are all functions of the same process p. . We arrive then at a very surprising conclusion: the homogeneity of beliefs of the investors, for whatever utility function they have, implies a complete homogeneity of behavior: their wealth, their consumption, the proportion of their wealth that they invest in the different assets at time t are all functions of the value of a given process pto which is log-normally distributed. Moreover, the marginal value of a unit of wealth in terms of utility and the marginal utility of consumption are equal for each individual and proportional to pto. The process p” plays here the role of an index of the market, which gives all the necessary information to all individuals on the state of the market and on their optimal decisions. The empirical relevance of this result can be questioned as well as the underlying model. First of all, uncertainty in stock market comes in great part from the heterogeneity of the beliefs of the stockholders. Moreover, different investors have access to different information. Conversely, identity and homogeneity of beliefs implies such an homogeneity of behavior that the uncertainty on others’ behavior is in great part cancelled. We see from this example the reason why homogeneity of beliefs implying homogeneity of behavior, this community of behavior tends to reduce uncertainty and makes beliefs more homogeneous again. In other words, at the level of the investor, this allows complete separation of market analysis on the one hand, and decision making on the other hand. An “invisible hand” makes all the investors who have the same beliefs follow the same score, even if everyone maximizes his own utility function for himself. Conversely, if two different investors “follow” the same process p. , while having possibly different log-normal representations of the price system, then for any i, they will have the same (ai - r)/oa . Figure 4 is a representation of the points (ffi, No). GROWTH AND OPTIMAL INTERTEMPORAL ALLOCATION 249 FIGURE 4 This means that both investors have the same instantaneous conditional market line on the market i (see [9, Chap. 51). This corresponds precisely to the homogeneity of beliefs. 2. A Growth Model We consider here a very simple one sector model with uncertainty. We assume that one factor is capital, called k, which can produce a given good through a production functionf(k). sf(k) dt is invested during interval dt and (1 - s)f(k) dt is consumed. Depreciation rate is 6. There is an uncertainty in the accumulation process, that is the conditional standard deviation of the increment will be a(k, sf(k)). Formally, we represent the accumulation process by: dk = @f(k) - Sk) df + a(k, sf(k)) . dfl, (4.15) k(0) = k, . The problem is to maximize the expected discounted intertemporal e-Ot U(( 1 ESW utility: (4.16) s>.fO) 4 0 with U concave in its argument and U’(O) = +co. transformed Hamiltonian. We have: c@ = U((1 - s) f(k)) + M-(k) Let us write the - Sk) + Wk, &W. (4.17) If we maximize &’ in s with 0 < s < 1, we get s(-V(C) + p + Hq) = 0.5 (4.18) Moreover, d’ = - t(1 - 8) f’(k) WC> + psf’(k) - I-J@ + p) + H(uK + s+‘(k) cq] dt + H * d/3 + dM. 5 01 and ok are the partial derivatives of o. (4.19) 250 JEAN-MICHEL BISMUT We can rewrite Eqs. (4.18) and (4.19), assuming s > 0: u‘(C) d’ = {-(P = P + HOI , (4.20) + H~I) f’(k) - Ha, + ~(6 + p)} dt + H . d/9 + dM. Let us try to interpret these relations. If we think of an exchange procedure between consumer and producer -by remembering that what is not consumed is invested, the first equality means that the consumer will consume up to the point where the marginal utility of his consumption equals the expected marginal value of capital in terms of utility minus the marginal risk of investment valued at its cost, because one unit of consumption less means not only one unit of investment more, but also-in the case where u increases with I-some risk which is taken in the accumulation process. At a time where the consumer is risk averter, that is when -H > 0 (the conditional covariance of dp and dk is negative), this will tend to make the consumer consume more than he would with the same p, when no risk is involved, because he fears capital losses appearing in the negative conditional covariance of dp dk. The consumer pays then “only” p + Hul to the producer. But the producer, as the consumer, bargains also on the risk market. If R is the cost of capital, the producer will have to pay (in expected value) Rk - Ho. He receives (p + Hq) by unit of production. If we maximize in k the expression (4.21) we get (p + Hu,)f’(k) + HQ - R = 0. (4.22) Then the profit rate in terms of the marginal value of capital p is: r = R/P = (1 + (H/P) UI) f’(k) + (H/P) uk (4.23) Relation (4.20) can then be written: (4.24) This is nothing else than the neoclassical relation between interest rate p, net rate of return r - 6, and expected inflation rate E(dp/p dt). How can we, starting in an uncertain “bargain” between consumption and investment come to a relation which is characteristic of certainty or perfect capital markets, as established in [S] ? It is precisely through the creation of a supplementary market, which is the risk market. GROWTH AND OPTIMAL INTERTEMPORAL ALLOCATION 251 To make things clearer, if --His the cost of risk, the consumer computes the expected marginal price of consumption, which will become now a “real” price thanks to the introduction of the risk market: this real price is p + HaI . It is the maximum price he is ready to “pay” to the producer. Now for a production f(k) the producer receives the sure quantity (p + Hu,)f(k). But his “expected profit” which is changed into a real profit through the risk market is (p + Ho,)f(k) + Ho(k, I) - Rk, which he maximizes.6 We have then the following system of markets. The consumer buys C at price p and sells o to the producer at its price -H. He equals marginal utility to the marginal real cost. The producer sells his whole productionf(k) at a price equal to the marginal utility of consumption U’(C) and buys u at his cost -H from the consumer, as well as capital at cost R. This makes perfectly clear that the dynamics of production and consumption involves a transmission of uncertainty for a decision made at a time t to all the future. Time decentralization allows us to reduce the transmissions to an instantaneous transaction between production and consumption in which precisely uncertainty is traded at its cost. In particular -Hq is the risk premium that the consumer receives because, as a consumer, he will suffer from the uncertainty he might himself create now for the next period. If we comparef’(k) and r, in (4.23), we have f’(k) = r - (H/P) 0~ 1 + W/P) The instantaneous risk premium, ur : that isf’(k) (4.25) - r is equal to (4.26) A Worked Example We assume f(k) = bk with b> 0 a(& k) = Bk U(C) = ((C)l-V)/(l (4.27) - a). 8 Let us insist again here that ,3 is not necessarily a Brownian motion, and does not have necessarily continuous paths and that the representation of o as a standard deviation term is done for commodity. 252 JEAN-MICHJZL BISMUT We define s by s = 1 - Clbk. We must have O<S<l. The system can then be written dk = sbk * dt + 8k * d,$ k(0) = k, . (4.28) We want to maximize, for z, belonging to [O, l] ~“((1 - s) bk)l-” dt. (4.29) But k can be written: k, = k, exp ( jO’ sb du - $Pt + e/3). (4.30) Then: {k#-v = (k,, exp s,‘sb du)‘-’ geyl - exp({- exp(e(1 - V) p - tez(l - v)Z t) - 0) + ie2(1- 7~)“)t). (4.31) Let us define k’ by: dk’ = sbk’ dt, (4.32) k’(0) = k, . If we admit that s is deterministic, e+((l = (3% s: we have then: - S) bk))l-” dt exp[-{p + $32v(l - a)} t]((l - s) bk’)l-v dt. (4.33) But if b(1 - v) < p + (l/2) e2v(l - u) < b, (4.34) we know how to solve the optimization problem defined by Relations (4.32) and (4.33) for s deterministic, because if we define y by: y = p + (i/2) ew - 74, it is precisely the problem treated in [4] [Chap. 11, p. 3771. (4.35) GROWTH AND OPTIMAL INTERTEMPORAL 253 ALLOCATION If p’ is the dual variable associated to the deterministic one will check that the process problem, p = p’(exp @3 - (l/2) Pt}+, is precisely the dual process associated to the problem of control under uncertainty. The deterministic solution found for the second problem is then the solution for the first problem. Even if consumption, which is (1 - s) bk is random, it is remarkable that s has been found by solving a deterministic process with a higher discount rate p + (l/2) &J(l - U). (4.37) However, the interpretation of @%(l - u) as a risk premium must be done carefully: if one had chosen C instead of s as a control variable, no result of this sort could have been found, because the optimal C is random. But even though a result given in [4] has an easy interpretation: it says that the time of transition t, to full consumption, that is s = 0, is defined by (4.38) @MU - exp[-O - 4) = 1. By checking that tz decreases with y, we see that the addition of (l/2) 8%(1 - u) tends to make the consumer consume sooner, out of fear of insufficient time and too much uncertainty to enjoy consumption. But absolute risk aversion of U is v/C, and relative risk aversion is V. Risk aversion increases then with u, although ~(1 - u) does not. This proves at least that under uncertainty, there is some discrepancy between instantaneous risk aversion and intertemporal risk aversion. However, one can check that H is given on H = -pue. The instantaneous risk premium is then, by (4.26), a&. This instantaneous risk premium increases then with instantaneous risk aversion. 3. A Model with Death Time We assume that the final time T has a probability by he”* dt (t 3 0) and that fi = 0. It is known that in this case, to maximize e--gtU(C) dt 642/10/2-g distribution given 254 JEAN-MICHEL is equivalent to the maximization BISMUT of +m E e-fDtA)tU(C) dt. s0 (4.40) This can be easily “translated” in our framework. Actually, the dual variable associated to the problem (4.39) is given by: dp = (-&%“/ax) dt + pp dt + dM. (4.41) PT = 0 At time T, dM will be precisely the jump -pr- , because at this time, the marginal value of capital goes abruptly from pr- to 0. MT - MT- is then equal to -pT- . But it can then easily be proved that on t < T, one has: dM, = hp, . (4.42) Then Relation (4.41) can be written on f < T: dp = -(aX/ax) dt + (p + 4 P df. (4.43) PT = 0. The quantity hp dt is then “added” to compensate exactly in expected value the possible return to 0 of p.’ Our dual process p is then nothing else that the modified dual process relative to Relation (4.40) for t < T and 0 for t > T. One sees here how a new and sudden arrival of information obliges to a reevaluation of predictions. Again, we come back to a deterministic problem by increasing the certain discount rate of a given risk premium. V. UNCERTAINTY AND TIME PREFERENCES In the worked example of Section IV.2, we have seen that the certain discount rate corresponding to the optimal strategy in an uncertain growth model was not increasing monotonically with instantaneous risk aversion. This leads us naturally to say that the effects of risk are generally ambiguous in an intertemporal framework. Although our analysis will be only tentative, we now try to shed some light on the reasons for this ambiguity. To simplify the exposition we 1 Obviously, simple methods exist for this example which is, however, illustrative. GROWTH AND OPTIMAL INTERTEMPORAL ALLOCATION 255 suppose that the choice is between certain present consumption and future uncertain consumption. Uncertainty in the future tends to decrease the value of future benefits, because of risk aversion. There are then two effects. An “income” effect which tends to make the investor poorer even now; he tends then to diminish his present consumption, in order to save for future uncertain periods. In terms of risk theory, this is an attitude of “insurance” against future uncertainty by present saving. A “substitution effect” which tends to substitute present benefits for future benefits, related to the irreversibility of the sequence present-future, in complete difference with the deterministic case, where the distinction between present and future is arbitrary: the conceptual impossibility of the investor going back from future to present tends to make him give more importance to present benefits. These two effects-the insurance effect and the irreversibility effectare both present and work in opposition: the first tends to raise the preference for future, the second to lower it. The certain equivalent rate of discount is then neither necessarily a monotonic function of risk aversion, nor a monotonic function of the size of future risks: these last risks, when they are very large, may tend to increase the “insurance” effect, and to lower the certain equivalent rate of discount. It is then not necessarily true that reduction of future risks tends to reduce the certain private rate of discount. The old attitude of gold accumulation in French families, known as the effect of “bas de lame” corresponds to a private rate of return of zero. The reduction of future uncertainties and a decrease in risk aversion can lead, in this case, to a larger social rate of discount. Although these “unnatural” aspects of risk taking may be considered pathological and irrelevant to present situations, we are led to think that the effects of risk on private and social attitudes are still to be explored. CONCLUSION It may be useful to summarize briefly some of the ideas developed in this paper. The first idea was to solve the problem of intertemporal allocation 256 JEAN-MICHEL BISMUT of risks, by introducing a sequence of risk markets, on which present risk could be traded versus its expected future consequences. This may be done by using a new maximum principle developed in [2] and [3]. By applying this model to the stock-market model of Merton [7], we found the existence of a stock market index: we were then led to question the relevance of any model which assumes homogeneity of beliefs, and more generally, to ask in what sense social uncertainty is generated precisely by the heterogeneity of beliefs. A stochastic growth model led us also to establish relations generalizing the corresponding neoclassical relations between marginal productivity of capital, expected inflation rate, depreciation rate, interest rate, and cost of risk. In particular, we found that there is no clear-cut relation between instantaneous risk aversion and the certain equivalent intertemporal discount rate. To explain this surprising result, we have introduced two effects coming from uncertainty of the future: the “insurance” effect and the “irreversibility” effect. Although the reasonings and interpretations are not always easy, conclusions are illuminating and helpful in understanding the problem of intertemporal optimal allocation of risks. The application of such an approach to multisector models can be very helpful, because then a difficult question is raised: In what sector must one allocate resources to be at the optimum? In particular, when we have two sectors, a government sector and a private sector, what is the influence of risk on investment in the public and private sectors? There are in these cases several risk markets, and choice of sectors and investments is done through a comparison of their respective conditional mean and standard deviation. Marginal productivities of private and public capital will then reflect the different expectations and risks associated to investments in these two sectors. REFERENCES 1. K. J. ARROW AND M. KURZ, “Public investment, the rate of return, and optimal fiscal policy,” Johns Hopkins Press, Baltimore, 1970. 2. J. M. BLSMLIT, Analyse Convexe et Probabilites, Doctoral Dissertation, Faculte des Sciences de Paris, 1973. 3. J. M. BISMUT, Conjugate convex fonctions in optimal stochastic control. J. MutA. and Appl., 44 (November 1973), 384404. TER ANLI A. R. DOBELL, “Mathematical Theories of Economic Growth,” 4. E. BW Macmillan, New York, 1970. 5. R. D~RFMAN, An Economic Interpretation of Optimal Control Theory, A.E.R. (December 1969), pp. 817-831. 6. E. FAMA, Efficient capital markets. J. Finance, (May 1970), pp. 383-417. GROWTH AND OPTIMAL INTERTFMPORAL ALLOCATION 257 7. R. C. MERTON, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory 3 (1971), 373-413. 8. P. A. SAMUELSON, Some Aspects of the Pure Theory of Capital, Quart. J. Econ. LI (May, 1937), 469-496. 9. W. F. SHARPE, Portfolio theory and capital markets, McGraw-Hill, New York, 1970. 642/10/z-10